Here \(\mathbf {p}=(p_1,p_2,\dots )\)  and \(\tilde{\mathbf {p}}=(\tilde{p}_1,\tilde{p}_2,\dots )\)  are two infinite sets of
variables and \(s_{(\alpha |\beta )}\)  denotes the Schur polynomial [1]} in the power sum variables labeled
by a partition \((\alpha |\beta )\)  written in the Frobenius coordinates \(\alpha =(\alpha _1,\dots ,\alpha _\kappa )\) 
and \(\beta =(\beta _1,\dots ,\beta _\kappa )\) .
This is also the simplest example of the Toda lattice (TL) [2]} tau functions [3]},[4]}
where the variables \(\mathbf {p}\)  and \(\tilde{\bf {p}}\)  play the role of the so-called higher times.
For higher times,
other designations are sometimes used, for example, \(n x_n\)  instead of \(p_n\) , the sign can be also different.
To fix the correspondence we write the Appendix. It is written in
form of series in the Schur functions as it was worked out in [5]}, [6]}.
The straightforward generalization of this expression to the \(\textsc {N}\) -component Toda lattice (\(\textsc {N}\) -TL) yields
\(\tau _o(\mathbf {p}^{1},\tilde{\bf {p}}^{1},\dots ,\mathbf {p}^{\textsc {N}},\tilde{\bf {p}}^{\textsc {N}})=1+ \sum _{\kappa > 0}\sum _{\alpha ^1_1>\cdots >\alpha ^1_\kappa ,\atop \beta ^1_1>\cdots >\beta ^1_\kappa }\cdots \sum _{\alpha ^\textsc {N}_1>\cdots >\alpha ^\textsc {N}_\kappa ,\atop \beta ^\textsc {N}_1>\cdots >\beta ^\textsc {N}_\kappa }\,\prod _{1\le i\le \textsc {N}}^{\circlearrowleft }s_{(\alpha ^{i}|\beta ^{i})}(\mathbf {p}^{i})s_{(\beta ^{i}|\alpha ^{i+1})}(\tilde{\bf {p}}^{i})\) 
